Chapter 16. Logic Programming. Topics. Logic Programming. Logic Programming Paradigm

Size: px

Start display at page:

Download "Chapter 16. Logic Programming. Topics. Logic Programming. Logic Programming Paradigm"

Kathryn Walters
5 years ago
Views:

1 Topics Chapter 16 Logic Programming Introduction Predicate Propositions Clausal Form Horn 2 Logic Programming Paradigm AKA Declarative Paradigm The programmer Declares the goal of the computation (specification of results are stated). Does not declare a detailed algorithm by which these goals are to be achieved. Application domain Database design Natural language processing Artificial Intelligence Automatic theorem proving Example language: Prolog Logic Programming Instead of providing implementation, execute specification. Relieves the programmer of specifying the implementation. Express programs in a form of symbolic logic. Declarative specification: Given an element x and a list L, to prove that x is in L, proceed as follows: Prove thatlis [x]. Otherwise, split L into L 1 and L 2 and prove one of the following: x is in L 1 or x is in L Logic Programming Less effort to write, but implementation may be very inefficient. Requires that the execution engine be more complex. Use a logical inferencing (INFERENCE ENGINE) process to produce results Introduction to Predicate Symbolic logic can be used for the basic needs of formal logic: Express propositions Express relationships between propositions Describe how new propositions can be inferred from other propositions Particular form of symbolic logic used for logic programming is called first-order predicate calculus 5 6 1

2 Introduction to Predicate Proposition: a logical statement that may or may not be true. Consists of objects and relationships of objects to each other. Can either assert truth ( john speaks Russian ) or query existing knowledge base ( does john speak Russian ). Can contain variables, which can become bound speaks(x,russian). Introduction to Predicate Example (English statements Predicate ) 0 is a natural number natural(0). 2 is a natural number natural(2). For all x, if x is a natural number, then so is the successor of x. For all x, natural(x) natural(successor(x)). -1 is a natural number natural(-1). 7 8 Introduction to Predicate First and third logical statements are axioms for the natural numbers. Statements that are assumed to be true and from which all true statements about natural numbers can be proved. Second logical statement can be proved from the previous axioms. 2 = successor(successor(0)). natural(0) natural(successor(successor(0)). Fourth logical statement cannot be proved from the axioms and so can be assumed to be false. 9 Predicate : statements Predicate calculus classifies the different parts of statements as: 1. Constants. These are usually number or names. Sometimes they are called atoms, since they cannot be broken down into subparts. Example: 1, 0, true, false 2. Predicates. These are names for functions that are true or false, like Boolean functions in a program. Can take any number of arguments. Example: the predicate natural takes one argument. 10 Predicate : statements 3. Functions. Predicate calculus distinguishes between functions that are true or false these are predicates and all other functions, which represent non-boolean values. Example: successor 4. Variables. Variables stand for as yet unspecified quantities. Example: x 5. Connectives. These include the operations and, or, and not, just like the operations on Boolean data in programming languages. Additional connectives are implication and equivalence Predicate : table of connectives Name Negation Conjunction Disjunction Equivalence Implication Symbol Example a a b a b a b a b a b not a a and b a or b Meaning a is equivalent to b a implies b b implies a Notes True if a is false; otherwise false True if a and b are both true True if either a or b (or both) is true True if a and b are both true or both false Logically equivalent to a b

3 Predicate : connectives By convention, negation has highest precedence. Conjunctions, disjunctions, and equivalence have higher precedence than implication (in that order). Example: p q r s t is equivalent to ((p (q r)) (( s) t)) Predicate : quantifiers 6. Quantifiers. These are operations that introduce variables. Universal Quantifier: for all Existential Quantifier: there exists A variable introduced by a quantifier is said to be bound by the quantifier. It is possible for variables also to be free (not bound by any quantifier). Quantifiers have higher precedence than any of the operators Name Universal Existential Predicate : table of quantifiers Symbol Example X P X P For all X, P is true Meaning There exists a value of X such that P is true Predicate : quantifiers Examples: x(speaks(x,russian)) True if everyone on the planet speaks Russian; false otherwise. x(speaks(x,russian)) True if at least one person on the planet speaks Russian; false otherwise. x y(speaks(x,y)) True if every person x speaks some language y; false otherwise. x y(speaks(x,y)) True if at least one person on the planet speaks every language y; false otherwise Predicate : statements 7. Punctuation Symbols. These include left and right parentheses the coma, and the period. Parentheses can be left out based on common conventions about the precedence of connectives. Arguments to predicates and functions can only be terms, that is, combinations of variables, constants, and functions. Terms cannot contain predicates, quantifiers, or connectives. Predicate : examples prime(n) True if the integer value of n is a prime number. 0 x + y True if the real sum of x and y is nonnegative. speaks(x,y) True if the person x speaks language y. 0 x + x 1 True if x is between 0 and 1, inclusive. speaks(x,russian) speaks(y,russian) talkswith(x,y) True if the fact that both x and y speak Russian implies that x talks with y x( literate(x) ( writes(x) y(reads(x,y) book(y)))) True if every illiterate person x does not write and has not read any book y

4 Predicate : tautologies Tautologies: Propositions that are true for all possible values of their variables. Example: q q Predicates that are true for some particular assignment of values to their variables are called satisfiable. Example: speaks(x,russian) If at least one person in the planet speaks Russian. Predicates that are true for all possible assignments of values to their variables are valid. Example: even(y) odd(y) It is true for all integers 19 Propositions: Summary Objects in propositions are represented by simple terms: either constants or variables Constant: a symbol that represents an object Variable: a symbol that can represent different objects at different times Different from variables in imperative languages 20 Propositions: Summary Simplest propositions are called atomic propositions which consist of compound terms A compound term is composed of two parts Functor: function symbol that names the relationship. Ordered list of parameters (tuple) Propositions: Summary Examples: student(jon) like(seth, OSX) like(nick, windows) like(jim, linux) Propositions can be stated in two forms: Fact: proposition is assumed to be true Query: truth of proposition is to be determined Compound proposition: Have two or more atomic propositions Propositions are connected by operators Clausal Form Clausal Form Problem of predicate calculus: Too many ways to state the same thing Solution: use a standard form for propositions All propositions can be expressed in clausal form: Antecedent: right side of proposition. Consequent: left side of the proposition. likes(bob,mary) likes(bob,redheads) redhead(mary). B 1 B 2 B n A 1 A 2 A m means if all the As are true, then at least one B is true Characteristics of clausal form: Existential quantifiers are not required. Universal quantifiers are implicit with use of variables. No operator other than conjunctions and disjunctions. consequent antecedent A proposition with zero or one terms in the consequent is called a Horn clause

5 Horn A Horn clause has a head h, which is a predicate, and a body, which is a list of predicates p 1,p 2, p n p 1,p 2, p n h body head In a Horn clause the head is true if every predicate of the body is true (simultaneously). Horn : facts and queries Fact: a Horn clause without body. They are called headless Horn clauses. h or just h It means that h is always true. Example: mammal(human). Query: a Horn clause without a head. The opposite of a fact. Example: mammal(human) From Predicates to Horn There is a limited correspondence between Horn clauses and predicates. Horn clauses can be written equivalently as a predicate HC: snowing(c) precipitation(c),freezing(c). PC: snowing(c) precipitation(c) freezing(c). Not all predicates can be translated into Horn clauses. Properties of Predicate Logic Expressions Property Commutativity Associativity Distributivity Idempotence Identity demorgan Implication Quantification Meaning p q q p (p q) r p (q r) p q r (p q) (p r) p p p p p true (p q ) p q p q p q x P(x) x P(x) p q q p (p q) r p (q r) p (q r) (p q) (p r) p p p p p false (p q ) p q x P(x) x P(x) From Predicates to Horn Six-step procedure that will, whenever possible, translate a predicate p into a Horn clause. 1. Eliminate implications from p, using the implication property. 2. Move negation inward in p, using the demorgan and quantification properties, so that only individual terms are negated. 3. Eliminate existential quantifiers from p, using a technique called skolemization. Here, the existentially quantified variable is replaced by a unique constant. For example, the expression xp(x) is replaced by P(c), where c is an arbitrarily chosen constant in the domain of x. From Predicates to Horn 4. Move all universal quantifiers to the beginning of p; as long as there are no naming conflicts, this step does not change the meaning of p. Assuming that all variables are universally quantified, we can drop the quantifiers without changing the meaning of the predicates. 5. Use the distributive, associative, and commutative properties to convert p to conjunctive normal form. In this form, the conjunction and disjunction operators are nested no more than two level deep, with conjunctions at the highest level

6 From Predicates to Horn 6. Convert the embedded disjunctions to implications, using the implication property. If each of these implications has a single term on its right, then each can be rewritten as a series of Horn clauses equivalent to p. Example: x( literate(x) ( writes(x) y(reads(x,y) book(y)))) Example: x(literate(x) reads(x) writes(x)) 31 6

Introduction to Sets and Logic (MATH 1190)

Introduction to Sets and Logic (MATH 1190) Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition